Randomized Cunningham Numbers in Cryptography

In this research we study primality testing of arbitrary integers via number theory randomized algorithms and optimization theories. Concerning fundamentals of modern cryptography we focus on the cryptanalysis cryptosystems and RSA keys which are widely used in secure communications e.g. banking systems and other platforms with an online security. For a given n-bit integer N our consideration is realized as a decision problem viz. an optimized algorithm rendering the output YES if N is a prime and NO otherwise. In order to design such an algorithm we begin by examining essential ingredients from the number theory namely divisibility modular arithmetic integer representations distributions of primes primality testing algorithms greatest common divisor least common multiplication pseudoprimes Np-hard discrete logarithm problem residues and others. Hereby we give special attention to the congruence relations Chinese remainder theorem and Fermat''s little theorem towards the optimal primality testing of an integer. Finally we anticipate optimized characterizations of Cunningham numbers in the light of randomization theory and their applications to cryptography.